4.1.20. Knak: segmented power law transmission model

The knak model is used to model the transmission of any spectrum using a set of contiguous segments with a power-law transmission at each segment.

This component can be useful for new instruments, in order to test large scale calibration errors (effective area errors for example), but other applications can also be made, of course. For example, if the spectrum of the source has an unknown continuum shape with a superimposed absorption model, it is a good trick to model the continuum by a power law, modify that by a knak model with adjustable or fixed transmission at some relevant energies, and then apply the absorption. An example of this last application can be found in Porquet et al. (2004).

The Transmission is given by:

T(E) = c_iE^{a_i} \quad \mathrm{if} \quad E_i < E < E_{i+1}

for each value of i between 0 and n, the number of grid points. The transmission is 1 for E<E_1 and E>E_2. Further, instead of using the constants c_i and a_i, we use instead the values of the transmission at E_i, i.e. T_i \equiv T(E_i) = c_i E_i^{\displaystyle{a_i}}. This allows for a continuous connection between neighbouring segments.

Finally, for historical reasons we use here a wavelength grid instead of an energy grid; the corresponding nodes \lambda_i should therefore be in strictly increasing order.

Warning

When applying this model, take care that at least one of the n transmission values is kept fixed (otherwise you may run the risk that your model is unconstrained, for example if the normalisation of the continuum is also a free parameter).

The parameters of the model are:

n : The number of grid points. Maximum value is 9.
w1 : Wavelength \lambda_1 (Å) of the first grid point
f1 : Transmission T(\lambda_1) at \lambda_1.
w2 : Wavelength \lambda_2 (Å) of the second grid point
f1 : Transmission T(\lambda_2) at \lambda_2.
w9 : Wavelength \lambda_9 (Å) of the last grid point
f9 : Transmission T(\lambda_9) at \lambda_9.
Note that if n<9, the values of T_i and \lambda_i will be ignored for i>n.

Recommended citation: Porquet et al. (2004).